Mathematics

Write the value of the following integral : $$\underset{-\pi / 2}{\overset{\pi/2}{\int}} \sin^5 \, x \, dx$$


ANSWER


SOLUTION
Let  $$f(x)=\sin^{5}x$$

$$f(-x)=\sin^{5}(-x)$$

$$f(-x)=-\sin^{5}x=-f(x)$$

So $$f(x)$$ is an odd function.

And for any odd function,
$$\int_{-a}^{a}f(x)\: dx=0$$

So, the value of the given definite integral is zero.
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